Answer
$\omega_{CD}=0~rad/s$
Work Step by Step
We can determine the required angular velocity as follows:
$sin\phi=\frac{x}{0.3}$
$\implies x=0.3sin\phi$
and $cos\phi=\frac{y}{0.3}$
$y=0.3cos\phi$
Thus, $tan\theta=\frac{x}{0.6-y}$
$\implies tan\theta=\frac{0.3sin\phi}{0.6-0.3cos\phi}$
$\implies 0.3sin\phi=tan\theta(0.6-0.3cos\phi)$
$\implies 0.3sin\phi=0.6tan\theta-0.3cos\phi tan\theta$
Differentiating with respect to time, we obtain:
$0.3cos\phi\frac{d\phi}{dt}=0.6sec^2\frac{d\theta}{dt}-0.3cos\phi sec^2\theta \frac{d\theta}{dt}+0.3sin\phi tan\theta \frac{d\phi}{dt}$
We plug in the known values to obtain:
$0.3cos60\omega_{AB}=0.6sec^2 30\omega_{CD}-0.3cos60 sec^2 30\omega_{CD}+0.3sin60tan30\omega_{AB}$
$0.3cos60(4)=0.6sec^2 30\omega_{CD}-0.3cos60sec^230\omega_{CD}+0.3sin60tan30(4)$
This simplifies to:
$\omega_{CD}=0rad/s$
